Fuzzy concept

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A fuzzy concept is a concept of which the meaningful content, value, or boundaries of application can vary considerably according to context or conditions, instead of being fixed once and for all.[1] This generally means the concept is vague, lacking a fixed, precise meaning, without however being meaningless altogether.[2] It has a meaning, or multiple meanings (it has different semantic associations). But these can become clearer only through further elaboration and specification, including a closer definition of the context in which they are used. Fuzzy concepts "lack clarity and are difficult to test or operationalize".[3]

In logic, fuzzy concepts are often regarded as concepts which in their application, or formally speaking, are neither completely true nor completely false, or which are partly true and partly false; they are ideas which require further elaboration, specification or qualification to understand their applicability (the conditions under which they truly make sense).

In mathematics and statistics, a fuzzy variable (such as "the temperature", "hot" or "cold") is a value which could lie in a probable range defined by quantitative limits or parameters, and which can be usefully described with imprecise categories (such as "high", "medium" or "low") using some kind of qualitative scale.

In mathematics and computer science, the gradations of applicable meaning of a fuzzy concept are described in terms of quantitative relationships defined by logical operators. Such an approach is sometimes called "degree-theoretic semantics" by logicians and philosophers,[4] but the more usual term is fuzzy logic or many-valued logic. The basic idea is, that a real number is assigned to each statement written in a language, within a range from 0 to 1, where 1 means that the statement is completely true, and 0 means that the statement is completely false, while values less than 1 but greater than 0 represent that the statements are "partly true", to a given, quantifiable extent. This makes its possible to analyze a distribution of statements for their truth-content, identify data patterns, make inferences and predictions, and model how processes operate.

Fuzzy reasoning (i.e., reasoning with graded concepts) has many practical uses.[5] It is nowadays widely used in the programming of vehicle and transport electronics, household appliances, video games, language filters, robotics, and various kinds of electronic equipment used for pattern recognition, surveying and monitoring (such as radars). Fuzzy reasoning is also used in artificial intelligence and virtual intelligence research.[6] "Fuzzy risk scores" are used by project managers and portfolio managers to express risk assessments.[7]

Origins and etymology[edit]

The intellectual origins of the idea of fuzzy logic have been traced to a diversity of famous and less well known thinkers including Plato, Georg Wilhelm Friedrich Hegel, Karl Marx, Friedrich Engels, Friedrich Nietzsche, Jan Łukasiewicz, Alfred Tarski, Stanisław Jaśkowski[8] and Donald Knuth.[9] However, usually the Iranian computer scientist Lotfi A. Zadeh is credited with inventing the specific idea of a "fuzzy concept" in his seminal 1965 paper on fuzzy sets, because he gave a formal mathematical presentation of the phenomenon which was widely accepted by scholars.[10] In fact, the German scholar Dieter Klaua also published a German-language paper on fuzzy sets in the same year,[11] but he used a different terminology (he referred to "many-valued sets").[12] Radim Belohlavek explains:

"There exists strong evidence, established in the 1970s in the psychology of concepts... that human concepts have a graded structure in that whether or not a concept applies to a given object is a matter of degree, rather than a yes-or-no question, and that people are capable of working with the degrees in a consistent way. This finding is intuitively quite appealing, because people say "this product is more or less good" or "to a certain degree, he is a good athlete", implying the graded structure of concepts. In his classic paper, Zadeh called the concepts with a graded structure fuzzy concepts and argued that these concepts are a rule rather than an exception when it comes to how people communicate knowledge. Moreover, he argued that to model such concepts mathematically is important for the tasks of control, decision making, pattern recognition, and the like. Zadeh proposed the notion of a fuzzy set that gave birth to the field of fuzzy logic..."[13]

Hence, a concept is regarded as "fuzzy" by logicians if:

  • defining characteristics of the concept apply to it "to a certain degree or extent" (or with a certain magnitude of likelihood)
  • or, the fuzzy concept itself consists of a fuzzy set.

The fact that a concept is fuzzy does not prevent its use in logical reasoning, it merely affects the type of reasoning which can be applied (see fuzzy logic).

The idea of fuzzy concepts was subsequently applied in the philosophical, sociological and linguistic analysis of human behaviour. In a 1973 paper, George Lakoff for example analyzed hedges in the interpretation of the meaning of categories.[14] Charles Ragin and others have applied the idea to sociological analysis.[15]

In a more general sociological or journalistic sense, a "fuzzy concept" has come to mean a concept which is meaningful but inexact, implying that it does not exhaustively or completely define the meaning of the phenomenon to which it refers - often because it is too abstract. To specify the relevant meaning more precisely, additional distinctions, conditions and/or qualifiers would be required. For example, in a handbook of sociology we find a statement such as "The theory of interaction rituals contains some gaps that need to be filled and some fuzzy concepts that need to be differentiated."[16] The idea is that if finer distinctions are introduced, then the fuzziness or vagueness would be eliminated.

The main reason why the term is now often used in describing human behaviour, is that human interaction has many characteristics which are difficult to quantify and measure precisely, among other things because they are interactive and reflexive (the observers and the observed mutually influence the meaning of events).[17] Those human characteristics can be usefully expressed only in an approximate way (see reflexivity (social theory)).[18] Newspaper stories frequently contain fuzzy concepts, which are readily understood and used, even although they are far from exact. Thus, many of the meanings which people ordinarily use to negotiate their way through life in reality turn out to be "fuzzy concepts". While people often do need to be exact about some things (e.g. money or time), many areas of their lives involve expressions which are far from exact.

Uncertainty[edit]

Fuzzy concepts can generate uncertainty because they are imprecise (especially if they refer to a process in motion, or a process of transformation where something is "in the process of turning into something else"). In that case, they do not provide a clear orientation for action or decision-making ("what does X really mean or imply?"); reducing fuzziness, perhaps by applying fuzzy logic, would generate more certainty.

However, this is not necessarily always so.[19] A concept, even although it is not fuzzy at all, and even though it is very exact, could equally well fail to capture the meaning of something adequately. That is, a concept can be very precise and exact, but not - or insufficiently - applicable or relevant in the situation to which it refers. In this sense, a definition can be "very precise", but "miss the point" altogether.

A fuzzy concept may indeed provide more security, because it provides a meaning for something when an exact concept is unavailable - which is better than not being able to denote it at all. A concept such as God, although not easily definable, for instance can provide security to the believer.

Language[edit]

Ordinary language, which uses symbolic conventions and associations which are often not logical, inherently contains many fuzzy concepts - "knowing what you mean" in this case depends on knowing the context or being familiar with the way in which a term is normally used, or what it is associated with. This can be easily verified for instance by consulting a dictionary, a thesaurus or an encyclopedia which show the multiple meanings of words, or by observing the behaviours involved in ordinary relationships which rely on mutually understood meanings. Bertrand Russell regarded language as intrinsically vague.[20]

To communicate, receive or convey a message, an individual somehow has to bridge his own intended meaning and the meanings which are understood by others, i.e., the message has to be conveyed in a way that it will be socially understood, preferably in the intended manner. Thus, people might state: "you have to say it in a way that I understand".

This may be done instinctively, habitually or unconsciously, but it usually involves a choice of terms, assumptions or symbols whose meanings may often not be completely fixed, but which depend among other things on how the receiver of the message responds to it, or the context. In this sense, meaning is often "negotiated" or "interactive" (or, more cynically, manipulated). This gives rise to many fuzzy concepts.

But even using ordinary set theory and binary logic to reason something out, logicians have discovered that it is possible to generate statements which are logically speaking not completely true or imply a paradox,[21] even although in other respects they conform to logical rules.

Psychology[edit]

The origin of fuzzy concepts is partly due to the fact that the human brain does not operate like a computer (see also Chinese room).[22]

  • While computers use strict binary logic gates, the brain does not; i.e., it is capable of making all kinds of neural associations according to all kinds of ordering principles (or fairly chaotically) in associative patterns which are not logical but nevertheless meaningful. For example, a work of art can be meaningful without being logical. A pattern can be regular and non-arbitrary, hence meaningful, without it being possible to describe it completely or exhaustively in formal-logical terms.
  • Something can be meaningful although we cannot name it, or we might only be able to name it and nothing else.
  • The human brain can also interpret the same phenomenon in several different but interacting frames of reference, at the same time, or in quick succession, without there necessarily being an explicit logical connection between the frames.

In part, fuzzy concepts arise also because learning or the growth of understanding involves a transition from a vague awareness, which cannot orient behaviour greatly, to clearer insight, which can orient behaviour.

Some logicians argue that fuzzy concepts are a necessary consequence of the reality that any kind of distinction we might like to draw has limits of application. As a certain level of generality, a distinction works fine. But if we pursued its application in a very exact and rigorous manner, or overextend its application, it appears that the distinction simply does not apply in some areas or contexts, or that we cannot fully specify how it should be drawn. An analogy might be that zooming a telescope, camera, or microscope in and out reveals that a pattern which is sharply focused at a certain distance disappears at another distance (or becomes blurry). Faced with any large, complex and continually changing phenomenon, any short statement made about that phenomenon is likely to be "fuzzy", i.e., it is meaningful, but - strictly speaking - incorrect and imprecise. It will not really do justice to the reality of what is happening with the phenomenon. A correct, precise statement would require a lot of elaborations and qualifiers. Nevertheless, the "fuzzy" description turns out to be a useful shorthand that saves a lot of time in communicating what is going on ("you know what I mean").

In psychophysics it has been discovered that the perceptual distinctions we draw in the mind are often more sharply defined than they are in the real world. Thus, the brain actually tends to "sharpen up" our perceptions of differences in the external world. Between black and white, we are able to detect only a limited number of shades of gray, or colour gradations. If there are more gradations and transitions in reality, than our conceptual or perceptual distinctions can capture, then it could be argued that how those distinctions will actually apply, must necessarily become vaguer at some point. If, for example, one wants to count and quantify distinct objects using numbers, one needs to be able to distinguish between those separate objects, but if this is difficult or impossible, then, although this may not invalidate a quantitative procedure as such, quantification is not really possible in practice; at best, we may be able to assume or infer indirectly a certain distribution of quantities that must be there.

Finally, in interacting with the external world, the human mind may often encounter new, or partly new phenomena or relationships which cannot (yet) be sharply defined given the background knowledge available, and by known distinctions, associations or generalizations.

"Crisis management plans cannot be put 'on the fly' after the crisis occurs. At the outset, information is often vague, even contradictory. Events move so quickly that decision makers experience a sense of loss of control. Often denial sets in, and managers unintentionally cut off information flow about the situation" - L. Paul Bremer, "Corporate governance and crisis management", in: Directors & Boards, Winter 2002

It also can be argued that fuzzy concepts are generated by a certain sort of lifestyle or way of working which evades definite distinctions, makes them impossible or inoperable, or which is in some way chaotic. To obtain concepts which are not fuzzy, it must be possible to test out their application in some way. But in the absence of any relevant clear distinctions, or when everything is "in a state of flux" or in transition, it may not be possible to do so, so that the amount of fuzziness increases.

Occurrence of fuzzy concepts in different areas[edit]

Fuzzy concepts often play a role in the creative process of forming new concepts to understand something. In the most primitive sense, this can be observed in infants who, through practical experience, learn to identify, distinguish and generalise the correct application of a concept, and relate it to other concepts.[23]

However, fuzzy concepts may also occur in scientific, journalistic, programming and philosophical activity, when a thinker is in the process of clarifying and defining a newly emerging concept which is based on distinctions which, for one reason or another, cannot (yet) be more exactly specified or validated. Fuzzy concepts are often used to denote complex phenomena, or to describe something which is developing and changing, which might involve shedding some old meanings and acquiring new ones.

  • In politics, it can be highly important and problematic how exactly a conceptual distinction is drawn, or indeed whether a distinction is drawn at all; distinctions used in administration may be deliberately sharpened, or kept fuzzy, due to some political motive or power relationship. Politicians may be deliberately vague about some things, and very clear and explicit about others; if there is information that proves their case, they become very precise, but if the information doesn't prove their case, they become vague or saying nothing. The "fuzzy area" can also refer simply to a residual number of cases which cannot be allocated to a known and identifiable group, class or set.
  • In administration and accounting, fuzziness problems of interpretation and boundary problems can arise, because it is not clear to what category exactly a case, item, transaction or piece of information belongs. In principle, each case, event or item must be allocated to the correct category in a procedure, but it may be, that it is difficult to make the appropriate or relevant distinctions.
  • In translation work, fuzzy concepts are analyzed for the purpose of good translation. A concept in one language may not have quite the same meaning or significance in another language, or it may not be feasible to translate it literally, or at all. Some languages have concepts which do not exist in another language, raising the problem of how one would most easily render their meaning. In computer-assisted translation, a technique called fuzzy matching is used to find the most likely translation of a piece of text, using previous translated texts as a basis.
  • In information services fuzzy concepts are frequently encountered because a customer or client asks a question about something which could be interpreted in many different ways, or, a document is transmitted of a type or meaning which cannot be easily allocated to a known type or category, or to a known procedure. It might take considerable inquiry to "place" the information, or establish in what framework it should be understood.
  • In the legal system, it is essential that rules are interpreted and applied in a standard way, so that the same cases and the same circumstances are treated equally. Otherwise one would be accused of arbitrariness, which would not serve the interests of justice. Consequently, lawmakers aim to devise definitions and categories which are sufficiently precise that they are not open to different interpretations. For this purpose, it is critically important to remove fuzziness, and differences of interpretation are typically resolved through a court ruling based on evidence. Alternatively, some other procedure is devised which permits the correct distinction to be discovered and made.
  • In medical diagnosis, the assessment of what the symptoms of a patient are often cannot be very exactly specified, since there are many possible qualitative and quantitative gradations in severity, incidence or frequency that could occur. Different symptoms may also overlap to some extent. These gradations can be difficult to measure, and so the medical professionals use approximate "fuzzy" categories in their judgement of a medical condition or a patient's state of health. Although it may not be exact, the diagnosis is often useful enough for treatment purposes.
  • In statistical research, it is an aim to measure the magnitudes of phenomena. For this purpose, phenomena have to be grouped and categorized so that distinct and discrete counting units can be defined. It must be possible to allocate all observations to mutually exclusive categories so that they are properly quantifiable. Survey observations do not spontaneously transform themselves into countable data; they have to be identified, categorized and classified in such a way, that identical observations can be grouped together, and that observations are not counted twice or more. Again, for this purpose it is a requirement that the concepts used are exactly defined, and not fuzzy. There could be a margin of error, but the amount of error must be kept within tolerable limits, and preferably its magnitude should be known.
  • In hypnotherapy, fuzzy language is deliberately used for the purpose of trance induction. Hypnotic suggestions are often couched in a somewhat vague, general or ambiguous language requiring interpretation by the subject. The intention is to distract and shift the conscious awareness of the subject away from external reality to his own internal state. In response to the somewhat confusing signals he gets, the awareness of the subject spontaneously tends to withdraw inward, in search of understanding or escape.[24]
  • In biology, protein complexes with multiple structural forms are called fuzzy complexes. The different conformations can result in different, even opposite functions. The conformational ensemble is modulated by the environmental conditions. Post-translational modifications or alternative splicing can also impact the ensemble and thereby affinity or specificity of interactions.
  • In theology an attempt is made to define more precisely the meaning of spiritual concepts, which refer to how human beings construct the meaning of human existence, and, often, the relationship people have with a supernatural world. Many spiritual concepts and beliefs are fuzzy, to the extent that, although abstract, they often have a highly personalized meaning, or involve personal interpretation of a type that is not easy to define in a cut-and-dried way. A similar situation occurs in psychotherapy. The Dutch theologian Kees de Groot has explored the imprecise notion that psychotherapy is like an "implicit religion", defined as a "fuzzy concept" (it all depends on what one means by "psychotherapy" and "religion").[25]
  • In meteorology, where changes and effects of complex interactions in the atmosphere are studied, the weather reports often use fuzzy expressions indicating a broad trend, likelihood or level. The main reason is that the forecast can rarely be totally exact for any given location.
  • In phenomenology which studies the structure of subjective experience, an important insight is that how someone experiences something can be influenced both by the influence of the thing being experienced itself, but also by how the person responds to it.[26] Thus, the actual experience the person has, is shaped by an "interactive object-subject relationship". To describe this experience, fuzzy categories are often necessary, since it is often impossible to predict or describe with great exactitude what the interaction will be, and how it is experienced.

It could be argued that many concepts used fairly universally in daily life (e.g. "love" or "God" or "health" or "social") are inherently or intrinsically fuzzy concepts, to the extent that their meaning can never be completely and exactly specified with logical operators or objective terms, and can have multiple interpretations, which are in part exclusively subjective. Yet despite this limitation, such concepts are not meaningless. People keep using the concepts, even if they are difficult to define precisely.

It may also be possible to specify one personal meaning for the concept, without however placing restrictions on a different use of the concept in other contexts (as when, for example, one says "this is what I mean by X" in contrast to other possible meanings). In ordinary speech, concepts may sometimes also be uttered purely randomly; for example a child may repeat the same idea in completely unrelated contexts, or an expletive term may be uttered arbitrarily. A feeling or sense is conveyed, without it being fully clear what it is about.

Fuzzy concepts can be used deliberately to create ambiguity and vagueness, as an evasive tactic, or to bridge what would otherwise be immediately recognized as a contradiction of terms. They might be used to indicate that there is definitely a connection between two things, without giving a complete specification of what the connection is, for some or other reason. This could be due to a failure or refusal to be more precise. But it could also could be a prologue to a more exact formulation of a concept, or a better understanding.

Fuzzy concepts could also simply be a practical method to describe something of which a complete description would be an unmanageably large undertaking, or very time-consuming; thus, a simplified indication of what is at issue is regarded as sufficient, although it is not exact. There is also such a thing as an "economy of distinctions", meaning that it is not helpful or efficient to use more detailed definitions than are really necessary for a given purpose. The provision of "too many details" could be disorienting and confusing, instead of being enlightening, while a fuzzy term might be sufficient to provide an orientation. The reason for using fuzzy concepts can therefore be purely pragmatic, if it is not feasible for practical purposes to provide "all the details" about the meaning of a shared symbol or sign. Thus people might say "I realize this is not exact, but you know what I mean" - they assume practically that stating all the details is not required for the purpose of the communication.

Analysis[edit]

In mathematical logic, computer programming, philosophy and linguistics fuzzy concepts can be analyzed and defined more accurately or comprehensively, by describing or modelling the concepts using the terms of fuzzy logic. More generally, clarification techniques can be used such as:

  • concretizing the concept by finding specific examples, illustrations or cases to which it applies.
  • specifying a range of conditions to which the concept applies (for example, in computer programming of a procedure).
  • classifying or categorizing all or most cases or uses to which the concept applies (taxonomy).
  • probing the assumptions on which a concept is based, or which are associated with its use (Critical thought).
  • identifying operational rules for the use of the concept, which cover all or most cases.
  • allocating different applications of the concept to different but related sets (e.g. using Boolean logic).
  • examining how probable it is that the concept applies, statistically or intuitively (Probability theory).
  • examining the distribution or distributional frequency of (possibly different) uses of the concept (statistics).
  • some other kind of measure or scale of the degree to which the concept applies.
  • specifying a series of logical operators (an inferential system or algorithm) which captures all or most cases to which the concept applies.
  • mapping or graphing the applications of the concept using some basic parameters.
  • applying a meta-language which includes fuzzy concepts in a more inclusive categorical system which is not fuzzy.
  • reducing or restating fuzzy concepts in terms which are simpler or similar, and which are not fuzzy or less fuzzy.
  • relating the fuzzy concept to other concepts which are not fuzzy or less fuzzy, or simply by replacing the fuzzy concept altogether with another, alternative concept which is not fuzzy yet "works exactly the same way".
An operationalization diagram, one method of clarifing fuzzy concepts.

In this way, we can obtain a more exact understanding of the use of a fuzzy concept, and possibly decrease the amount of fuzziness. It may not be possible to specify all the possible meanings or applications of a concept completely and exhaustively, but if it is possible to capture the majority of them, statistically or otherwise, this may be useful enough for practical purposes.

A process of defuzzification is said to occur, when fuzzy concepts can be logically described in terms of (the relationships between) fuzzy sets, which makes it possible to define variations in the meaning or applicability of concepts as quantities. Effectively, qualitative differences may then be described more precisely as quantitative variations or quantitative variability (assigning a numerical value then denotes the magnitude of variation).

The difficulty that can occur in judging the fuzziness of a concept can be illustrated with the question "Is this one of those?". If it is not possible to clearly answer this question, that could be because "this" (the object) is itself fuzzy and evades definition, or because "one of those" (the concept of the object) is fuzzy and inadequately defined.

Thus, the source of fuzziness may be in the nature of the reality being dealt with, the concepts used to interpret it, or the way in which the two are being related by a person. It may be that the personal meanings which people attach to something are quite clear to the persons themselves, but that it is not possible to communicate those meanings to others except as fuzzy concepts.

See also[edit]

References[edit]

  1. ^ Susan Haack, Deviant logic, fuzzy logic: beyond the formalism. Chicago: University of Chicago Press, 1996.
  2. ^ Richard Dietz & Sebastiano Moruzzi (eds.), Cuts and clouds. Vagueness, Its Nature, and Its Logic. Oxford University Press, 2009.
  3. ^ Ann Markusen, "Fuzzy Concepts, Scanty Evidence, Policy Distance: The Case for Rigour and Policy Relevance in Critical Regional Studies." In: Regional Studies, Volume 37, Issue 6-7, 2003, pp. 701-717.
  4. ^ Roy T. Cook, A dictionary of philosophical logic. Edinburgh University Press, 2009, p. 84.
  5. ^ Kazuo Tanaka, An Introduction to Fuzzy Logic for Practical Applications. Springer, 1996; Constantin Zopounidis, Panos M. Pardalos & George Baourakis, Fuzzy Sets in Management, Economics and Marketing. Singapore; World Scientific Publishing Co. 2001.
  6. ^ Stosberg, Mark (16 December 1996). "The Role of Fuzziness in Artifical Intelligence". Minds and Machines. Retrieved 19 April 2013. 
  7. ^ Irem Dikmen, M. Talat Birgonal and Sedat Han, "Using fuzzy risk assessment to rate cost overrun risk in international construction projects." International Journal of Project Management, Vol. 25 No. 5, July 2007, pp. 494-505.
  8. ^ Susan Haack notes that Stanisław Jaśkowski provided axiomatizations of many-valued logics in: Jaśkowski, "On the rules of supposition in formal logic. Studia Logica No. 1, 1934.[1] See Susan Haack, Philosophy of Logics. Cambridge University Press, 1978, p. 205.
  9. ^ Priyanka Kaushal, Neeraj Mohan and Parvinder S. Sandhu, "Relevancy of Fuzzy Concept in Mathematics". International Journal of Innovation, Management and Technology, Vol. 1, No. 3, August 2010.[2]
  10. ^ Lotfi A. Zadeh, "Fuzzy sets". In: Information and Control, Vol. 8, June 1965, pp. 338–353.[3]
  11. ^ Dieter Klaua. "Über einen Ansatz zur mehrwertigen Mengenlehre". Monatsberichte der Deutschen Akademie der Wissenschaften (Berlin), Vol. 7: pp. 859-867, 1965.
  12. ^ Siegfried Gottwald, "Shaping the logic of fuzzy set theory". In: Cintula, Petr et al. (eds.), Witnessed years. Essays in honour of Petr Hájek. London: College Publications, 2009, pp. 193-208. [4]
  13. ^ Radim Belohlavek, "What is a fuzzy concept lattice? II", in: Sergei O. Kuznetsov et al. (eds.), Rough sets, fuzzy sets, data mining and granular computing. Berlin: Springer Verlag, 2011, pp. 19-20.[5]
  14. ^ George Lakoff, "Hedges: A Study in Meaning Criteria and the Logic of Fuzzy Concepts." Journal of Philosophical Logic, Vol. 2, 1973, pp. 458-508.[6]
  15. ^ Charles Ragin, Redesigning Social Inquiry: Fuzzy Sets and Beyond. University of Chicago Press, 2008. Shaomin Li, "Measuring the fuzziness of human thoughts: An application of fuzzy sets to sociological research". The Journal of Mathematical Sociology, Volume 14, Issue 1, 1989, pp. 67-84.
  16. ^ Jörg Rössel and Randall Collins, "Conflict theory and interaction rituals. The microfoundations of conflict theory." In: Jonathan H. Turner (ed.), Handbook of Sociological Theory. New York: Springer, 2001, p. 527.
  17. ^ Loïc Wacquant, "The fuzzy logic of practical sense." in: Pierre Bourdieu and Loïc Wacquant, An invitation to reflexive sociology. London: Polity Press, 1992, chapter I section 4.
  18. ^ Ph. Manning “Fuzzy Description: Discovery and Invention in Sociology”. In: History of the Human Sciences, Vol. 7, No. 1, 1994, pp. 117-23.
  19. ^ Masao Mukaidono, Fuzzy logic for beginners. Singapore: World Scientific Publishing, 2001.
  20. ^ Bertrand Russell, "Vagueness" (1923) in: Bertrand Russell Papers, Vol. 9, pp. 147–54. Nadine Faulkner, "Russell and vagueness." Journal of Bertrand Russell Studies, Summer 2003, pp. 43-63.
  21. ^ Patrick Hughes & George Brecht, Vicious Circles and Infinity. An anthology of Paradoxes. Penguin Books, 1978.
  22. ^ See further Radim Belohlavek & George J. Klir (eds.) Concepts and Fuzzy Logic. MIT Press, 2011. John R. Searle, "Minds, brains and programs". The behavioral and brain sciences, Vol. 3, No. 3, 1980, pp. 417-457.
  23. ^ Philip J. Kelman & Martha E. Arterberry, The cradle of knowledge: development of perception in infancy. Cambridge, Mass.: The MIT Press, 2000.
  24. ^ Ronald A. Havens (ed.), The wisdom of Milton H. Erickson, Volume I: hypnosis and hypnotherapy. New York: Irvington Publishers, 1992, p. 106. Joseph O'Connor & John Seymour (ed.), Introducing neuro-linguistic programming. London: Thorsons, 1995, p. 116f.
  25. ^ C.N. de Groot, "Sociology of religion looks at psychotherapy." Recherches sociologiques (Louvain-la-Neuve, Belgium), Vol. 29, No. 2, 1998, pp. 3-17 at p. 4.[7]
  26. ^ Michael Hammond, Jane Howarth and Russell keat, Understanding Phenomenology. Oxford: Blackwell, 1991.

External links[edit]

  • James F. Brule, Fuzzy systems tutorial [8]
  • "Fuzzy Logic", Stanford Encyclopedia of Philosophy [9]
  • "Vagueness", Stanford Encyclopedia of Philosophy [10]